Theorem tpssd 32794

Description: Deduction version of tpssi : An unordered triple of elements of a class is a subset of that class. (Contributed by Thierry Arnoux, 2-Nov-2025.)

Hypotheses

Ref	Expression
tpssd.1	⊢ (𝜑 → 𝐴 ∈ 𝐷)
tpssd.2	⊢ (𝜑 → 𝐵 ∈ 𝐷)
tpssd.3	⊢ (𝜑 → 𝐶 ∈ 𝐷)

Assertion

Ref	Expression
tpssd	⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Proof of Theorem tpssd

Step	Hyp	Ref	Expression
1		tpssd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
2		tpssd.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
3		tpssd.3	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐷)
4		tpssi 4799	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
5	1, 2, 3, 4	syl3anc 1394	1 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Syntax hints:   → wi 4   ∈ wcel 2145   ⊆ wss 3907  {ctp 4589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-un 3912  df-ss 3924  df-sn 4586  df-pr 4588  df-tp 4590

This theorem is referenced by:  constrlccllem  34060  constrcccllem  34061  cos9thpiminplylem2  34090